Lovász's conjecture on Hamiltonian paths in vertex-transitive graphs (1970)

Lovász's conjecture (1970). Every finite connected vertex-transitive graph has a Hamiltonian path.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-lovasz-conjecture,
  title        = {Lovász's conjecture on Hamiltonian paths in vertex-transitive graphs (1970)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/lovasz-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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The problem

The question

Lovász's conjecture (1970).

Every finite connected vertex-transitive graph has a Hamiltonian path.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.LovaszConjecture.

theorem lovasz_conjecture :
    ∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],
      G.Connected → IsVertexTransitive G → HasHamiltonianPath G

What counts as progress

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References

Wikipedia
[Lo70] Lovász, L. (1970). Problem 11, in Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf. 1969), Gordon and Breach, p. 497.
[Ba79] Babai, L. (1979). "Long cycles in vertex-transitive graphs." J. Graph Theory 3, pp. 301--304.
[CHM14] Christofides, D., Hladký, J. and Máthé, A. (2014). "Hamilton cycles in dense vertex-transitive graphs." J. Combin. Theory Ser. B 109, pp. 34--72. arXiv:1008.2193
[KM09] Kutnar, K. and Marušič, D. (2009). "Hamilton cycles and paths in vertex-transitive graphs — current directions." Discrete Math. 309, pp. 5491--5500.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.